To solve the differential equation \( \csc(y) \, dx + \sec^2(x) \, dy = 0 \), we can proceed as follows:

### Step 1: Separate the variables
We can rewrite the equation by moving all terms involving \(x\) to one side and all terms involving \(y\) to the other side:

\[
\csc(y) \, dx = -\sec^2(x) \, dy
\]

Now, divide both sides by \(\csc(y)\) and multiply both sides by \(dy\):

\[
\frac{dx}{\sec^2(x)} = -\frac{dy}{\csc(y)}
\]

### Step 2: Express in integrable form
Recalling that \(\sec^2(x) = \frac{1}{\cos^2(x)}\) and \(\csc(y) = \frac{1}{\sin(y)}\), the equation becomes:

\[
\cos^2(x) \, dx = -\sin(y) \, dy
\]

### Step 3: Integrate both sides
Now, integrate both sides:

\[
\int \cos^2(x) \, dx = -\int \sin(y) \, dy
\]

To integrate \( \cos^2(x) \), use the trigonometric identity:

\[
\cos^2(x) = \frac{1 + \cos(2x)}{2}
\]

Thus, the integral becomes:

\[
\int \frac{1 + \cos(2x)}{2} \, dx = \int \cos^2(x) \, dx
\]

Now integrate both sides:

\[
\frac{x}{2} + \frac{\sin(2x)}{4} = \text{Integration on the left side}
\]

\[
\int \cos^2(x) \, dx = \frac{x}{2} + \frac{\sin(2x)}{4} + C_1
\]

On the right side:

\[
\int \sin(y) \, dy = -\cos(y) + C_2
\]

So we have:

\[
\frac{x}{2} + \frac{\sin(2x)}{4} = -\cos(y) + C
\]

Where \(C = C_2 - C_1\) is the constant of integration.

### Final Solution
The general solution is:

\[
\frac{x}{2} + \frac{\sin(2x)}{4} + \cos(y) = C
\]

Where \(C\) is the constant of integration.

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Would you like any details or have any questions about this solution?

Here are 5 related questions that expand on this topic:
1. How would you solve the equation if different trigonometric functions were involved?
2. Can you explain the method of integrating trigonometric functions in more detail?
3. What are other common methods for solving first-order differential equations?
4. How do you handle separable differential equations where the variables are not initially separable?
5. Could you explain the role of constants of integration in solving differential equations?

**Tip:** When integrating trigonometric identities, remember that converting them into simpler forms (using identities) can often make the integration process more straightforward.

Learn how to solve the differential equation \( \csc(y) dx + \sec^2(x) dy = 0 \) with detailed steps and explanations. This problem involves differential equations, separable equations, and trigonometric identities for integration. Suitable for advanced college-level students and enthusiasts of differential equations.

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